Some Examples of Affine Monoid Algebras

Abstract

We construct a finitely generated monoid S with a zero element such that for every field K the Jacobson radical of the monoid algebra K[S] is a sum of nilpotent ideals but is not nilpotent. Moreover, the contracted monoid algebra K ₀[S] is a monomial algebra.

If K is a field of characteristic p > 0, then we construct a finitely presented group H _p such that the Jacobson radical J of the group algebra K[H _p ] is a sum of nilpotent ideals, but is not nilpotent. Moreover, K[H _p ]/J is a domain.

Key Words:

• Finitely presented algebra
• Jacobson radical
• Monoid algebra

2000 Mathematics Subject Classification:

• Primary 16N20, 16S15
• Secondary 16S34, 16S36, 20F05

Acknowledgments

Work supported by a MNiSW research grant N201 420539 (Poland).

Notes

Communicated by A. Smoktunowicz.